16.07.02 · inorgchem / solid-state

Crystal structures: close-packing, unit cells, and ionic radius ratios

stub3 tiersLean: nonepending prereqs

Anchor (Master): West — Solid State Chemistry and its Applications (1984)

Intuition Beginner

Imagine stacking oranges at a grocery store. You lay out the first layer so each orange sits in the hollow between the oranges below it. This is close-packing — arranging spheres so they fill the maximum possible space. Atoms in many metals and ionic solids pack the same way.

There are two close-packing patterns. Cubic close-packed (CCP), also called face-centred cubic (FCC), stacks layers in an ABCABC sequence. Hexagonal close-packed (HCP) stacks layers in an ABABAB sequence. Both fill 74% of the available space — the densest possible arrangement of equal spheres.

Ionic compounds like NaCl use a close-packed anion framework with cations sitting in the gaps (holes) between the larger anions. The ratio of cation radius to anion radius predicts which holes get filled and therefore which crystal structure forms. NaCl adopts the rock-salt structure (octahedral holes, coordination number 6). ZnS adopts either the zinc blende or wurtzite structure (tetrahedral holes, coordination number 4).

Visual Beginner

The difference between FCC and HCP packing is the stacking sequence of close-packed layers. In both cases each layer is identical — the variation comes from where each new layer sits relative to the two below it.

Worked example Beginner

Predict the structure type for MgO using the radius ratio rules.

Step 1: Look up the ionic radii. $r$ (Mg $^{2 +}$ ) = 72 pm, $r$ (O $^{2 -}$ ) = 140 pm.

Step 2: Calculate the radius ratio: $r_{+} / r_{-} = 72/140 = 0.514$ .

Step 3: Compare to the threshold ranges. The ratio 0.514 falls in the range 0.414–0.732, which predicts octahedral coordination (CN = 6) and therefore the NaCl (rock-salt) structure.

MgO does indeed crystallise in the rock-salt structure, with each Mg $^{2 +}$ surrounded by six O $^{2 -}$ and each O $^{2 -}$ surrounded by six Mg $^{2 +}$ . The lattice parameter is $a = 421$ pm.

Check your understanding Beginner

Formal definition Intermediate+

Close-packing geometry. A close-packed layer of equal spheres has each sphere touching six neighbours in the plane. When a second layer is placed, each sphere sits in a triangular depression, creating two types of interstitial hole: tetrahedral (one sphere directly above or below the triangle of three) and octahedral (two sets of three spheres from adjacent layers pointing toward each other).

The stacking sequence determines the structure type:

Stacking	Name	Lattice type	Holes per sphere
ABAB...	HCP	Hexagonal (P)	2 tetrahedral + 1 octahedral
ABCABC...	CCP/FCC	Cubic (F)	2 tetrahedral + 1 octahedral

Both arrangements have identical hole counts per sphere. The difference is purely in the layer registry.

Unit cell parameters. For an FCC lattice of spheres with radius $r$ :

Lattice parameter: $a = 2 r 2$
Nearest-neighbour distance: $d = 2 r$
Packing efficiency: $π / (32) = 0.7405$ (74.05%)

For HCP with ideal axial ratio ( $c / a = 8/3 \approx 1.633$ ):

Same packing efficiency: 74.05%
Unit cell contains 2 atoms (primitive) or 6 atoms (conventional hexagonal cell)

Radius ratio rules. The structure type is predicted from $r_{+} / r_{-}$ by comparing the cation to the hole size:

$r_{+} / r_{-}$ range	Coordination	Hole geometry	Structure type
< 0.155	2	Linear	Rare
0.155–0.225	3	Triangular	Rare for ionic
0.225–0.414	4	Tetrahedral	ZnS (zinc blende or wurtzite)
0.414–0.732	6	Octahedral	NaCl (rock salt)
> 0.732	8	Cubic	CsCl

The threshold values are derived from the geometry of each hole type. The tetrahedral threshold (0.225) comes from the condition that the cation touches all four anions in a tetrahedral hole. The octahedral threshold (0.414) comes from the cation touching all six anions. The cubic threshold (0.732) comes from the body-centred geometry.

Rock-salt (NaCl) structure. FCC anion lattice with all octahedral holes occupied by cations. Each ion has CN = 6. Unit cell contains 4 formula units. Lattice parameter $a = 2 (r_{+} + r_{-})$ .

Zinc blende (sphalerite) structure. FCC anion lattice with half the tetrahedral holes occupied (every other one). Each ion has CN = 4. Unit cell contains 4 formula units. This is the structure adopted by many compound semiconductors: GaAs, InP, CdTe.

Wurtzite structure. HCP anion lattice with half the tetrahedral holes occupied. Each ion has CN = 4. The stacking is ABAB rather than ABCABC. Wurtzite and zinc blende are polymorphs of ZnS — same composition, same coordination, different stacking sequence. The energy difference between them is small ( $\sim$ 10 kJ/mol), so both occur in nature.

Madelung constants for common structures:

Structure	Madelung constant $M$
NaCl (rock salt)	1.7476
CsCl	1.7627
ZnS (zinc blende)	1.6381
ZnS (wurtzite)	1.641
CaF $_{2}$ (fluorite)	2.519
TiO $_{2}$ (rutile)	2.408

The Madelung constant captures the total Coulomb energy per ion pair relative to a single nearest-neighbour interaction. Higher Madelung constants indicate more favourable electrostatic stabilisation, but the nearest-neighbour distance $r_{0}$ also changes with structure type, so the total lattice energy is a nontrivial balance between geometry and interionic spacing.

Counterexamples to common slips

The radius ratio rules are approximate, not exact. Many real compounds deviate from the predictions. KCl ( $r_{+} / r_{-} = 0.76$ ) should be CsCl-type by the rules but crystallises as NaCl-type at room temperature. Covalent bonding contributions, polarisability, and lattice energy competition all influence the actual structure adopted.
Wurtzite and zinc blende are not the same structure. They share the same composition (ZnS), the same coordination number (4), and the same nearest-neighbour geometry (tetrahedral). But the stacking sequence differs — ABAB for wurtzite versus ABCABC for zinc blende — giving different space groups and different physical properties (piezoelectricity in wurtzite, absent in zinc blende).
Close-packing does not require equal-sized spheres. The concept generalises to binary systems where the larger ion (usually the anion) forms the close-packed framework and the smaller ion occupies interstitial holes. The packing efficiency for the binary system depends on both radii, not just the framework ions alone.
The FCC lattice is not the same as the CCP arrangement. FCC describes the lattice type (a mathematical point set with face-centring translations). CCP describes the physical arrangement of spheres. They coincide for equal spheres, but the distinction matters when the basis contains more than one atom type.

Key theorem with proof Intermediate+

Proposition (Radius ratio threshold for octahedral coordination). The minimum radius ratio for a cation to touch all six anions in an octahedral hole is $r_{+} / r_{-} = 2 - 1 \approx 0.414$ .

Proof. In an octahedral hole, four anions form a square in the equatorial plane and two anions sit on the axis perpendicular to this plane through its centre. The cation sits at the centre.

The four equatorial anions are at distance $d$ from the centre, where $d$ is the nearest-neighbour distance. The anion radius is $r_{-}$ , so the distance from the centre of one equatorial anion to the centre of the adjacent equatorial anion (diagonal of the square) is $d 2$ .

For the anions to be in contact with each other (touching condition): $d 2 = 2 r_{-}$ . But $d = r_{+} + r_{-}$ (cation and anion touching along the axis). So $(r_{+} + r_{-}) 2 = 2 r_{-}$ .

Solving: $r_{+} 2 + r_{-} 2 = 2 r_{-}$ , giving $r_{+} 2 = r_{-} (2 - 2)$ , and $r_{+} / r_{-} = (2 - 2) / 2 = 2 - 1 \approx 0.414$ .

Below this ratio, the cation is too small to maintain contact with all six anions simultaneously. The structure becomes unstable against collapse to a lower-coordination arrangement (tetrahedral, CN = 4). $■$

Bridge. This geometric threshold connects to the lattice-energy framework: when $r_{+} / r_{-} < 0.414$ , the cation rattles in the octahedral hole and the structure can lower its energy by switching to tetrahedral coordination where the cation fits snugly. The radius-ratio rules are thus a geometric proxy for the energetic competition between structure types, which the Madelung-constant treatment in 16.07.01 formalises quantitatively.

Exercises Intermediate+

Exercise 3 (medium, numeric).

Calculate the ideal axial ratio $c / a$ for HCP. The unit cell has two atoms at $(0, 0, 0)$ and $(1/3, 2/3, 1/2)$ in fractional coordinates. The nearest-neighbour distance is $a$ . Derive $c$ from the condition that the atom at $(1/3, 2/3, 1/2)$ is equidistant from three atoms in the layer below.

Hint

The three nearest neighbours in the layer below form an equilateral triangle. The atom at $(1/3, 2/3, 1/2)$ sits above the centroid of this triangle at height $c /2$ . Use Pythagoras: the distance from the centroid to a vertex is $a / 3$ , and the nearest-neighbour distance is $a$ .

Answer

The distance from the atom at $(1/3, 2/3, 1/2)$ to the atom at $(0, 0, 0)$ must equal $a$ (all nearest-neighbour distances are equal in ideal HCP).

The in-plane component of this distance (from the centroid to a vertex of the equilateral triangle) is $a / 3$ .

The vertical component is $c /2$ .

By Pythagoras: $a^{2} = (a / 3)^{2} + (c /2)^{2} = a^{2} /3 + c^{2} /4$ .

$c^{2} /4 = a^{2} - a^{2} /3 = 2 a^{2} /3$

$c^{2} = 8 a^{2} /3$

$c / a = 8/3 = 2 2/3 \approx 1.633$

This is the ideal axial ratio. Real HCP metals deviate slightly (Mg: 1.624, Zn: 1.856, Cd: 1.886) due to non-spherical electron distributions.

Exercise 4 (medium, short-answer).

Explain why the zinc blende structure is favoured by compound semiconductors (GaAs, InP) while the rock-salt structure is favoured by alkali halides (NaCl, KBr). Address both the radius-ratio argument and the bonding argument.

Hint

Consider electronegativity differences, the directionality of covalent bonds, and the typical radius ratios for each compound class.

Answer

Radius-ratio argument: Alkali halides typically have $r_{+} / r_{-} > 0.414$ , predicting octahedral coordination (NaCl structure). Compound semiconductors like GaAs have smaller radius ratios ( $r_{+} / r_{-} \approx 0.41$ for GaAs), bordering the tetrahedral range.

Bonding argument: This is the deeper reason. Alkali halides are highly ionic (large electronegativity difference, $Δ χ \approx 2.3$ for NaCl). Ionic bonding is isotropic — it depends only on distance, not direction — so it favours the higher-coordination NaCl structure that maximises the number of cation-anion contacts.

Compound semiconductors have significant covalent character (smaller electronegativity difference, $Δ χ \approx 0.4$ for GaAs). Covalent bonds are directional, and tetrahedral coordination is the natural geometry for sp $^{3}$ -hybridised bonds. The directional bonding requirement overrides the radius-ratio prediction and stabilises the zinc blende structure with CN = 4.

Exercise 5 (hard, short-answer).

Derive the radius ratio threshold for tetrahedral coordination. Show that a cation in a tetrahedral hole touches all four anions when $r_{+} / r_{-} = 3/2 - 1 \approx 0.225$ .

Hint

In a regular tetrahedron with anions at the four vertices, the cation sits at the centroid. The distance from the centroid to a vertex is $3 a /4$ where $a$ is the edge length. Set the edge length to $2 r_{-}$ (anions touching) and the centroid-to-vertex distance to $r_{+} + r_{-}$ .

Answer

Consider a regular tetrahedron with four anions at the vertices. The edge length (anion-anion distance) is $2 r_{-}$ when anions are in contact.

The distance from the centroid to any vertex of a regular tetrahedron with edge $a$ is $6 a /4$ .

Setting $a = 2 r_{-}$ : the centroid-to-vertex distance is $6 \times 2 r_{-} /4 = r_{-} 6 /2$ .

For the cation to touch all four anions: $r_{+} + r_{-} = r_{-} 6 /2$ .

$r_{+} = r_{-} (6 /2 - 1) = r_{-} (3/2 - 1)$ .

$r_{+} / r_{-} = 3/2 - 1 = 6 /2 - 1 \approx 1.2247 - 1 = 0.225$ .

Below this ratio, the cation is too small to maintain contact with all four anions, and the tetrahedral arrangement becomes unstable. This threshold marks the lower bound for CN = 4 structures.

Exercise 6 (hard, short-answer).

The wurtzite and zinc blende polymorphs of ZnS differ only in their stacking sequence (ABAB vs ABCABC) yet wurtzite is piezoelectric while zinc blende is not (in its ideal form). Explain why, using the symmetry argument.

Hint

Piezoelectricity requires a polar axis — a unique direction in the crystal along which the positive and negative charge centres do not coincide. Which space group does each polymorph have?

Answer

Zinc blende has space group $F \overset{ˉ}{4} 3 m$ (No. 216), which is cubic. The [111] direction is polar (the bonds along this axis are all Zn-S in the same direction), but there are four equivalent $⟨ 111 ⟩$ directions related by the cubic symmetry. In the ideal structure these polar directions are symmetrically equivalent, and the net polarisation averages to zero. However, zinc blende can be piezoelectric under shear because $F \overset{ˉ}{4} 3 m$ lacks inversion symmetry — it is one of the few piezoelectric cubic structures.

Wurtzite has space group $P 6_{3} m c$ (No. 186), which is hexagonal and explicitly polar. The $c$ -axis is a unique polar direction: all Zn-S bonds along $c$ point the same way, and there is no symmetry operation that reverses this direction. The crystal has a spontaneous polarisation along $c$ even in the absence of strain. Under mechanical stress, the ionic displacements modify this polarisation, giving a strong piezoelectric response.

The key distinction: wurtzite has a single polar axis (hexagonal symmetry), while zinc blende has multiple equivalent polar axes (cubic symmetry). Materials grown in the wurtzite phase — GaN, AlN, ZnO — are the basis of modern piezoelectric and pyroelectric devices precisely because of this symmetry property.

Polymorphism, phase transitions, and complex structure types Master

Polymorphism in MX compounds. The existence of multiple structure types for the same composition is widespread. ZnS crystallises as both zinc blende (sphalerite) and wurtzite. TiO $_{2}$ exists as rutile, anatase, and brookite. SiO $_{2}$ has quartz, cristobalite, tridymite, coesite, and stishovite. The thermodynamically stable polymorph at a given temperature and pressure is the one with the lowest Gibbs free energy $G = H - T S$ .

The enthalpy difference between polymorphs is often small — a few kJ/mol — because the coordination environment and bond lengths are similar. The entropy difference can tip the balance: higher-symmetry polymorphs generally have higher entropy and become stable at elevated temperature. The zinc blende to wurtzite transition in ZnS occurs at approximately 1020 degrees C, driven by the vibrational entropy advantage of the hexagonal phase.

Pressure-induced transitions favour denser structures with higher coordination. CsCl ( $r_{+} / r_{-} = 0.93$ ) crystallises in the CsCl structure (CN = 8) at room temperature, but many compounds that are NaCl-type at ambient conditions transform to CsCl-type under pressure. RbCl, KCl, and NaBr all undergo this transition, reflecting the fact that the ionic radius decreases under compression and the effective radius ratio shifts.

Perovskite structure (ABO $_{3}$ ). The perovskite structure is one of the most important structure types in solid-state chemistry and materials science. The ideal perovskite (e.g., SrTiO $_{3}$ ) has a cubic unit cell with the A cation at the corners, the B cation at the body centre, and oxygens at the face centres. The B cation sits in an octahedron of six oxygens, and the A cation sits in a cuboctahedral cavity formed by twelve oxygens.

The Goldschmidt tolerance factor predicts whether a given ABO $_{3}$ composition will form a stable perovskite:

t = \frac{r _{A} + r _{O}}{2 ( r _{B} + r _{O} )}

For ideal cubic perovskite, $t \approx 1.0$ . Perovskites form when $0.8 < t < 1.05$ . When $t < 0.8$ , the A cation is too small for the cuboctahedral cavity and the structure distorts to ilmenite (FeTiO $_{3}$ ) or other structure types. When $t > 1.05$ , the B cation is too small and the structure may adopt the hexagonal perovskite or other arrangements.

Perovskites display a remarkable range of properties: ferroelectricity (BaTiO $_{3}$ ), superconductivity (YBa $_{2}$ Cu $_{3}$ O $_{7 - x}$ ), colossal magnetoresistance (La $_{1 - x}$ Sr $_{x}$ MnO $_{3}$ ), photovoltaic activity (CH $_{3}$ NH $_{3}$ PbI $_{3}$ ), and catalytic activity. This versatility arises from the structural flexibility of the perovskite framework — tilting and distortion of the BO $_{6}$ octahedra accommodate a wide range of A and B cation sizes without changing the basic topology.

Spinel structure (AB $_{2}$ O $_{4}$ ). The spinel structure has 32 oxygens in a cubic close-packed arrangement with 8 tetrahedral holes occupied by A cations and 16 octahedral holes occupied by B cations. The unit cell contains 8 formula units. The archetypal spinel is MgAl $_{2}$ O $_{4}$ .

In a normal spinel, the divalent A cations occupy tetrahedral sites and the trivalent B cations occupy octahedral sites: [A] $^{t e t}$ [B $_{2}$ ] $^{oc t}$ O $_{4}$ . In an inverse spinel, the divalent cations swap to octahedral sites and half the trivalent cations move to tetrahedral sites: [B] $^{t e t}$ [AB] $^{oc t}$ O $_{4}$ . Magnetite (Fe $_{3}$ O $_{4}$ ) is an inverse spinel: [Fe $^{3 +}$ ] $^{t e t}$ [Fe $^{2 +}$ Fe $^{3 +}$ ] $^{oc t}$ O $_{4}$ .

The preference for normal vs. inverse arrangement is governed by the competition between crystal-field stabilisation energy (CFSE) and Madelung energy. Transition-metal ions with large CFSE in octahedral sites (Cr $^{3 +}$ , d $^{3}$ ) strongly favour octahedral coordination and drive the spinel toward inverse arrangement when they are the B cation.

Zeolite frameworks. Zeolites are aluminosilicate minerals with open framework structures built from corner-sharing SiO $_{4}$ and AlO $_{4}$ tetrahedra. The frameworks contain channels and cages of molecular dimensions (3–10 Angstroms), giving zeolites their characteristic ability to selectively adsorb molecules based on size and shape — the basis of molecular sieving.

Over 250 distinct zeolite framework types are known, classified by their topology (the connectivity of tetrahedral nodes, independent of the specific T-atoms). The most common are FAU (faujasite, used in fluid catalytic cracking), MFI (ZSM-5, used in petrochemical synthesis), and LTA (zeolite A, used in detergents and gas separation).

The framework composition determines the charge balance: each AlO $_{4}$ tetrahedron introduces a formal negative charge that must be compensated by an extra-framework cation (Na $^{+}$ , K $^{+}$ , Ca $^{2 +}$ ). These exchangeable cations are the basis of zeolite ion-exchange applications (water softening, radioactive waste treatment). The combination of uniform pore size, high surface area, and tuneable acidity makes zeolites among the most important industrial catalysts.

X-ray diffraction and Bragg's law. Crystal structures are determined experimentally by X-ray diffraction. The condition for constructive interference from crystal planes with spacing $d$ is Bragg's law:

nλ = 2 d sin θ

where $λ$ is the X-ray wavelength, $θ$ is the angle of incidence, and $n$ is the order of diffraction. The set of $d$ -spacings and their intensities uniquely fingerprints the crystal structure. Powder diffraction (Debye-Scherrer method) identifies unknown phases by matching the diffraction pattern to a database. Single-crystal diffraction provides the full atomic coordinates, including thermal parameters.

The structure factor $F_{hk l}$ determines the intensity of each reflection:

F_{hk l} = j \sum f_{j} exp [2 π i (h x_{j} + k y_{j} + l z_{j})]

where $f_{j}$ is the atomic scattering factor of atom $j$ at position $(x_{j}, y_{j}, z_{j})$ and $(h, k, l)$ are the Miller indices. Systematic absences (zero-intensity reflections) arise from lattice centring and screw axes, providing direct information about the space group. The phase problem — the fact that diffraction measures $∣ F_{hk l} ∣^{2}$ but not the phase of $F_{hk l}$ — is the central challenge of crystallography, solved by Patterson methods, direct methods, or increasingly by machine-learning approaches.

Connections Master

Solid-state chemistry overview 16.07.01. This unit deepens the crystal-structure vocabulary introduced in 16.07.01, moving from the qualitative description of ionic, metallic, and covalent solids to the quantitative geometry of close-packing and structure-type prediction. The radius-ratio rules and Madelung constants developed here connect directly to the lattice-energy framework and band-theory treatment in the prerequisite unit.
Coordination chemistry and crystal field theory 16.03.01. The tetrahedral and octahedral coordination geometries that dominate crystal structures are the same geometries that determine d-orbital splitting in coordination complexes. The preference of Cr $^{3 +}$ for octahedral sites in spinels mirrors the large octahedral CFSE of d $^{3}$ ions studied in coordination chemistry. The structure-type prediction from ionic radii parallels the spectrochemical-series arguments for ligand-field stabilisation.
Periodic trends 16.01.01. Ionic radii are periodic properties. The radius-ratio rules translate periodic trends into structural predictions: as cation size increases down a group, the radius ratio increases and the favoured structure type shifts from zinc blende (CN = 4) to rock salt (CN = 6) to CsCl (CN = 8). This is observable in the halides of a single metal (e.g., LiI is 4-coordinate while LiF is 6-coordinate, despite the same cation) and in the halides of different metals with a common anion.
Perovskites and energy materials 16.07.03 pending. The perovskite structure introduced here is the structural basis for the defect chemistry and non-stoichiometry explored in 16.07.03. Oxygen vacancies in perovskites enable ionic conduction in solid-oxide fuel cells, while the mixed-valence framework supports electronic conduction in electrodes.

Historical notes Master

The study of crystal structures began with the 1912 discovery of X-ray diffraction by Max von Laue, who showed that X-rays are diffracted by the periodic array of atoms in a crystal. Walter Friedrich and Paul Knipping carried out the experimental demonstration, producing the first diffraction photograph of a copper sulfate crystal. von Laue received the Nobel Prize in 1914.

W. H. Bragg and W. L. Bragg developed the mathematical framework in 1913. Bragg's law ( $nλ = 2 d sin θ$ ) converted the diffraction pattern into a tool for determining atomic positions. The Braggs solved the structures of NaCl, KCl, diamond, CaF $_{2}$ , and pyrites in rapid succession, establishing that crystals are periodic arrays of atoms rather than continuous matter. W. L. Bragg received the Nobel Prize in 1915 at age 25.

Linus Pauling published a series of papers in 1927–1929 establishing the ionic radius concept and the radius-ratio rules, drawing on the growing body of crystal-structure data from X-ray diffraction. Pauling's ionic radii, derived from a self-consistent analysis of interatomic distances across many crystal structures, remained the standard set for decades. The competing set of ionic radii by Ahrens (1952) and the "crystal radii" of Shannon and Prewitt (1969, revised 1976) refined the values but retained Pauling's framework.

The classification of close-packing types was formalised by William Barlow in the 1890s, before X-ray diffraction existed. Barlow deduced the FCC and HCP arrangements from macroscopic crystal symmetry and density arguments alone — a remarkable achievement of pure geometric reasoning confirmed decades later by diffraction experiments.

The concept of polymorphism was known to mineralogists from the distinct crystal habits of SiO $_{2}$ polymorphs (quartz, tridymite, cristobalite) long before atomic-level understanding. The thermodynamic framework for polymorphic stability — the Gibbs free energy, the role of entropy in stabilising high-temperature phases, and the Clapeyron equation for pressure-temperature phase boundaries — was applied systematically to solid-state phase transitions by Carl Wagner and Arnold Muan in the 1950s.

The perovskite structure was first identified in the mineral CaTiO $_{3}$ by Gustav Rose in 1839 and named after the Russian mineralogist Lev Perovski. Its importance grew through the 20th century: the discovery of ferroelectricity in BaTiO $_{3}$ (1945), the high-temperature superconductors based on perovskite-like cuprates (Bednorz and Muller, 1986, Nobel Prize 1987), and the organometal halide perovskite photovoltaics (Kojima et al., 2009) established perovskites as one of the most studied structure types in materials science.

Bibliography Master

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